Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 6 - Inverse Circular Functions and Trigonometric Equations - Section 6.3 Trigonometric Equations II - 6.3 Exercises: 50c

Answer

There will be about 10 hours of daylight on November 5th and February 3rd.

Work Step by Step

$h = \frac{35}{3}+\frac{7}{3}sin~\frac{2\pi~x}{365}$ We can find $x$ when $h=10$: $h = \frac{35}{3}+\frac{7}{3}sin~\frac{2\pi~x}{365}$ $3h = 35+7sin~\frac{2\pi~x}{365}$ $sin~\frac{2\pi~x}{365} = \frac{3h-35}{7}$ $sin~\frac{2\pi~x}{365} = \frac{(3)(10)-35}{7}$ $sin~\frac{2\pi~x}{365} = -\frac{5}{7}$ Note that $arcsin(-\frac{5}{7}) = -0.7956$ There will be two values for x, where $0 \leq x \lt 365$, such that $sin\frac{2\pi~x}{365} = -\frac{5}{7}$ These two values occur when $\frac{2\pi~x}{365} = \pi + 0.7956$ and $\frac{2\pi~x}{365} = 2\pi - 0.7956$. We can find the value of $x$ in both cases. We can find $x$ when $\frac{2\pi~x}{365} = \pi+0.7956$: $\frac{2\pi~x}{365} = \pi+0.7956$ $x = \frac{(\pi+0.7956)(365)}{2\pi}$ $x = 229~days$ We can find the date that is 229 days after March 21st: March 31st: 10 days April 30th: 30 days + 10 days = 40 days May 31st: 31 days + 40 days = 71 days June 30th: 30 days + 71 days = 101 days July 31st: 31 days + 101 days = 132 days August 31st: 31 days + 132 days = 163 days September 30th: 30 days + 163 days = 193 days October 31st: 31 days + 193 days = 224 days November 5th: 5 days + 224 days = 229 days We can find $x$ when $\frac{2\pi~x}{365} = 2\pi-0.7956$: $\frac{2\pi~x}{365} = 2\pi-0.7956$ $x = \frac{(2\pi-0.7956)(365)}{2\pi}$ $x = 319~days$ We can find the date that is 319 days after March 21st: March 31st: 10 days April 30th: 30 days + 10 days = 40 days May 31st: 31 days + 40 days = 71 days June 30th: 30 days + 71 days = 101 days July 31st: 31 days + 101 days = 132 days August 31st: 31 days + 132 days = 163 days September 30th: 30 days + 163 days = 193 days October 31st: 31 days + 193 days = 224 days November 30th: 30 days + 224 days = 254 days December 31st: 31 days + 254 days = 285 days January 31st: 31 days + 285 days = 316 days February 3rd: 3 days + 316 days = 319 days There will be about 10 hours of daylight on November 5th and February 3rd.
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